Proposition 7.5.2.6. Let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{QCat}$ be a (strictly commutative) diagram of $\infty $-categories. Then the homotopy limit $ \underset {\longleftarrow }{\mathrm{holim}}(\mathscr {F})$ is an $\infty $-category. Moreover, $ \underset {\longleftarrow }{\mathrm{holim}}(\mathscr {F} )$ can be identified with a limit of the diagram
\[ \operatorname{N}_{\bullet }^{\operatorname{hc}}( \mathscr {F}): \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{QCat}) = \operatorname{\mathcal{QC}} \]
in the $\infty $-category $\operatorname{\mathcal{QC}}$.